In answering this question, we must assume that the computing hardware you are using is of true "Dual Corded" design, in which each power cord connects internally to a totally separate power supply. In a true "Dual Corded" device, the only thing that should be common to the two power cords is the Safety Grounding Conductor that connects to the computing device chassis. Unfortunately, there have been some "fly by night" products on the market, thankfully rare, which have gone so far as to actually have the "Dual Cords" spliced together inside the equipment and connected to only a single power supply. This is illegitimate, illegal, dangerous, and obviously completely unethical. You should have no concerns about major name products, but if you buy some interesting, off-brand "garage shop" device, perhaps you should look inside before plugging it in, because the answer that follows doesn't apply to stuff built this way.
Understand that the purpose of the power supply inside any computing device is to convert line voltage alternating current (AC) to the low voltage direct current (DC) required to run the computing circuitry. Therefore, the two independent power supplies, each connected to a different incoming AC line, completely isolate one AC line from the other. It is only on the DC side that power is paralleled, where positive and negative are clearly defined and "phase" is no longer an issue. Furthermore, DC paralleling is generally done via isolation diodes so that the two supplies "load share" and neither supply can back-feed and affect the other. Therefore, you should be able to operate any truly "Dual Corded, Dual Power Supply" device from any two power sources. Once could even be the utility company AC line and the other a local generator, with neither having any reference to the other. So long as both sources are within the operating voltage range of the computing device, each is of sufficient current capacity, and the entire system is properly grounded, there should be no concern. (Grounding is most often the thing that gets done wrong, and that's worth closely examining in any complex power system.)
Paralleling actual power sources, such as two generators or two UPS's, can only be done through proper paralleling gear. In simple terms, the paralleling gear keeps the two or more sources in phase synchronization, and also provides isolation against back-feeds and fault currents so that neither of the power sources "see" current from the other source. But if you simply connect two sources together, without regard to phase or anything else, then yes, you're certainly going to have major problems, on top of being contrary to code.
This was first published in September 2005